35.1.4 Some caveats and comments

Each of the above mentioned schemes has been implemented in MOM so that zero horizontal background diffusion is required; i.e., the schemes are stable. This situation is in contrast to the Cox (1987) isoneutral diffusion scheme which required roughly 10⁶-10⁷ cm²/sec horizontal background diffusion; otherwise, the solution blew-up (Griff ies et al. 1998). In summary, the new diffusion scheme does the following:

1.

It produces a zero flux of locally referenced potential density. The Cox (1987) scheme did not respect this property, and this led to the instability of that scheme.

2.

It reduces tracer variance, and produces downgradient oriented tracer fluxes when considering a particular finite volume.

3.

It computes the best approximation to the neutral directions within the limitations of the discrete lattice.

4.

It requires zero background diffusion to remain stable; the Cox (1987) scheme blew-up without this diffusion.

Unfortunately, there is no constraint with the new diffusion or skew-diffusion schemes equivalent to the positive definiteness possessed by certain advection schemes, such as FCT (Section 31.6). As such, it is possible to realize unphysical tracer values even though the schemes are numerically stable. Experience has shown that most problems occur with passive tracers when employing isoneutral diffusion without skew-diffusion. Additionally, the problems occur most readily in regions of steep isoneutral slopes. The reason that steep sloped regions (say slopes greater than 1/100 or so) are most problematical relates to the inability to resolve these regions with grid aspect ratios typically on the order of 1/1000. Namely, the three components to the tracer flux may not properly balance to provide a strictly downgradient diffusion and cross-gradient skew-diffusion. The active tracers, as they are constrained to preserve the locally defined potential density, appear less problematic than the passive tracers. Also, depending on the value of the thickness diffusivity, GM90 can reduce the problems since it acts to reduce slopes. At this time, there are no feasible alternatives to the current methods of implementing these schemes in MOM which may overcome this difficulty with positive definiteness. The note by Beckers et al. (1998) provides some useful speculation.

The numerical problems with isoneutral mixing in steep sloped regions prompts one to assess how far the physics indicates that steep sloped regions need to preserve the along isoneutral nature of the diffusion. Measurements clearly indicate that the use of 10⁶-10⁷ cm²/sec globally applied horizontal background diffusion is not justified due to its huge dianeutral nature in regions of even modest slopes. However, in regions of steep slopes, it is arguable (e.g., Treguier et al. 1997) that the mesoscale eddy processes are transporting tracers across the steeply sloped mean density surfaces. Additionally, such slopes are typically associated with boundary layer processes, which are far from adiabatic. These two points indicate that a fair amount of horizontally aligned background diffusion might be warranted in steep sloped regions. As mentioned above, MOM allows for the use of such mixing in steep sloped regions (Section 34.1.9 provides further details). As might be expected, adding some form of horizontal diffusion in steep sloped regions ameliorates many of the numerical problems.

Another caveat concerns the ubiquitous problems with dispersive advection schemes. Upon stabilizing the isoneutral diffusion scheme, it is now possible to remove the horizontal background diffusion previously required with the Cox (1987) scheme. In so doing, however, one is exposed to problems with dispersive advection schemes. These problems are most apparent next to topography in regions where the tracers are aligned with isoneutrals, and so diffusive fluxes are weak. This ``Peclet grid noise'' problem is fundamental to dispersive advection. It is not a problem with the rotated diffusion.